ANOTHER FIVE QUESTIONS:

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1 No peaking!!!!! See if you can do the following: f 5 tan 6 sin 7 cos 8 sin 9 cos 5 e e Epress sin Power Series Epansion: d as a Power Series: Estimate sin Estimate MACLAURIN SERIES ANOTHER FIVE QUESTIONS: d with an error of less than... e using series, correct to three decimal places. Find the coefficient of f e. / in the Maclaurin series for 5 Estimate. e d, correct to three decimal places.

2 MACLAURIN SERIES (ANSWERS) - = n n AND n n =... n... n n n 6 9 n n = n 5 5 n n n n tan = n n n 6 5 n n n n sin =......! 5! n! n! n n n cos =......!! n! n! sin = n n n n n n n ! 5!!! n n cos 5 = !! n! n! n n n n n n n n e =......!!! n! n n! n n n e n =......!!! n! n n! n n n n ln = n n n ln = n n n... n n... n n 5 n n n n =... C...! 5! 7! 5! ! 5! 7! 5!.... e !!! / / / /6 /!!!! 6! n n n n. NOTE:. e d !! 5.

3 TAYLOR SERIES iv ''' f '' a f a f a Taylor: f a f ' a a a a a...!!! Question # (Calculator) Let f be a function that has derivatives of all orders for all real numbers. iv f 5, f ', f '', f '''. f, Assume,. Write the third-degree Taylor polynomial for f about. Write the fourth-degree Taylor polynomial for g, where g f. Write the third-degree Taylor polynomial for h, where h and use it to approimate., about f t dt, about f.. Let h be defined as in #. Either find the eact value of h or eplain why it can t be determined. Question # (Calculator) The Taylor series about 5 for a certain function f converges to of convergence. The nth derivative of f at 5is given by f n 5. Write the third-degree Taylor polynomial for f about 5.. Find the radius of convergence of the Taylor series for f about 5. Show that the sith-degree Taylor polynomial for f about 5 approimates than. Question # (No Calculator) f for all in the interval n n!, and f n 5. n f 6 with error less n n A function f is defined by f for all in the interval of n convergence of the given power series.. Find the interval of convergence for this power series. Show the work that leads to your answer. f. Find lim. Write the first three nonzero terms and the general term for an infinite series that represents f. Find the sum of the series determined in #. d.

4 Question #..8 P f 5 f. P f P g P f 5 P h t t dt t t t 6 6 h f tdt cannot be determined because!!! f ' 5, f '' 5, f '''' P f 6 6 a f t is known only for t Question #, n5 n n n 5 n n 5 n n n f n 5 lim lim 5. n! n n n n n n n The radius of convergence is. f 6 with the 6 th degree Taylor polynomial at 6 is less then the first omitted The error in approimating term in the series. f 6 P f,56 n n Question # n n n lim lim n n n n n n n n At, the series is Alt. Series lim Diverges n n n At, the series is Divergence Test Diverges n f / / 9 / / 9 / lim lim... lim n n n f d n n n a / / Geometric S n n r / /

5 VOLUME REVIEW Find the volume of the solid that results when the area of the region enclosed by y,, and y is revolved about the.. ais y. y ais. the line y. the line 5. the line y 6. the line 7. the line y 8. the line 6 9. the line y Find the volume of the solid that results when the area of the region enclosed by y,, and y. has cross sections perpendicular to the ais that are squares.. has cross sections perpendicular to the ais that are semi circles.. has cross sections perpendicular to the ais that are rectangles whose height is 5 times the length of its base in the region. y

6 . VOLUME REVIEW (ANSWERS) 58.6 (Washer) V d V ( y ) dy 8.5 (Washer).. V d 5. (Disk) ( ) 5.67 (Disk). V y dy V ( ) d.888 (Washer) 5. V ( ) ( y ) ( ) dy.95 (Washer) 6. ( ) ( ) 9.5 (Washer) V d 7. 6 ( ) 6.67 (Washer) 8. V y dy V 6 d.67 (Shell) 9... V ( ) d (Washer) ( ) 8 V d {Area = s } V d. {Area = r. V 5 d {Area = } 5s } Reminders: If calculator, make sure to go to AT LEAST three decimal places. This will most likely be a NON-CALCULATOR problem this year.

7 POLAR EQUATIONS Question # (Calculator) Let R be the region inside the graph of the polar curve r and outside the graph of the polar r sin curve. Sketch the two polar curves in the y plane provided below and shade the region R.. Find the intersection of the two curves.. Find the area of R. r sin.. Find the area of 5. Find the TOTAL AREA of the object. y Question # (No Calculator) Consider the polar curve r sin for.. In the y plane provided below, sketch the curve. Find the area of the region inside the curve.. Find the slope of the curve at the point where y

8 POLAR EQUATIONS (ANSWERS) GRAPH FOR PROBLEM # GRAPH FOR PROBLEM # 5 QUESTION # sin sin sin or Area r d sin d sin d sin d sin sin d sind cos d cos sin 8 Area r d sin d sin sin d sin cos d cos sin QUESTION # Area sin d cos 6 d cos 6 d sin 6 OR: 6 / / / / Area sin d 6 cos 6 d sin 6 6 d r cos sin cos 6cos cos sin sin d d 6cos cos sin sin d dy y r sin y sin sin 6 cos sin d dy d sin cos dy 6cos sin sin cos 6 d

9 Analyzing the Graph of a Derivative: PROBLEM #. For what value(s) of does f have a relative maimum? Why?. For what value(s) of does f have a relative minimum? Why?. On what intervals is the graph of f concave up? Why?. On what intervals is f increasing? Why? 5. For what value(s) of does f have an inflection point? Why? PROBLEM # The graph of a function f consists of a semicircle and two line segments as shown above. Let g be the function given by g f t dt. Find g().. For what value(s) of does g have a relative maimum? Why?. For what value(s) of does g have a relative minimum? Why?. For what value(s) of does g have an inflection point? Why? 5. Write an equation for the line tangent to the graph of g at = PROBLEM # The graph below shows f, the derivative of function f. The graph consists of two semi circles and one line segment. Horizontal tangents are located at = and = 8 and a vertical tangent is located at =.. On what intervals is f increasing? Justify your answer.. For what values of does f have a relative minimum? Justify.. On what intervals is f concave up? Justify.. For what values of is f undefined? 5. Identify the coordinates for all points of inflection of f. 6. For what value of does f reach its maimum value? Justify. 7. If f () = 5, find f().

10 PROBLEM # ANSWERS:. at, because f ' changes from positive to negative at.. at, because f ' changes from negative to positive at.. (,) and (,5) because f ' is increasing on these intervals (thus. (, ) and (,5) because f ' on these intervals. f '' ). 5. at =, = and =, because f '' changes signs at these values of.. PROBLEM # ANSWERS: g f t dt. At. At, because g ' f, because g ' f, because g'' f '. At and changes from positive to negative at. changes from negative to positive at. negative to positive at. g and g ' f y 5. changes from positive to negative at and changes from PROBLEM # ANSWERS:. (,) and (,) because f ' for these values of.. At because '. (,) (8,) and (,) because f '' for these values of. [Or, f ' is increasing for these values of.] f changes from negative to positive at.. At and 5. At and 8 6. At f is increasing on (,) [ because f is increasing on (,) [ On the interval (,), On the interval (,), Because 8 5, f ' ] and f is decreasing on (,) [ f ' ]. Thus, ma can occur at or at f decreases by: f increases by: f ' d 8 f ' d 5 f decreases by 8 5 on the interval (,). Thus, maimum occurs at. 7. f f f ' d f f f ' d 58 f ' ] and

11 PARAMETRIC EQUATIONS Question # (Calculator) An object is moving along a curve in the y plane has position, d dy cos t and sin t for t. At time dt dt. Write an equation for the line tangent to the curve at,5.. Find the speed of the object at time t.. Find the total distance traveled by the object over the time interval t.. Find the position of the object at time t Question # (No Calculator) A moving particle has position, t y t at time t with:,5. t, the object is at position t y t at time t. The position of the particle at time t is,6 and the velocity vector at any time t is given by, t t.. Find the acceleration vector at time t. Find the position of the particle at time t.. For what time t does the line tangent to the path of the particle at, t y t have a slope of 8?. The particle approaches a line at t. Find the slope of this line. Show work that leads to conclusion. Question # (Calculator) A particle moves in the y plane so that its position at any time t, t, is given by: t t ln t and y t sin t. Sketch the path of the particle in the y plane below. Indicate the direction of motion along the path.. At what time t, t, does t attain its minimum value? What is the position, t y t of the particle at this time?. At what time t, t, is the particle on the y ais? Find the speed and the acceleration vector of the particle at this time.

12 dy t d cost / PARAMETRIC EQUATIONS (ANSWERS) Question # sin dy sin 5.6 y d cos8 t AND Speed cos 8 9sin.75 cos t.95 and y 5 sin t.96 t y t y Distance cos t 9sin t dt.58 Question # '', '', '', '', t t 7 7 t, y t t C, t C,6, y C, C C, C 5 t t dt t t & y 6 6 t / dy t t t d t t t 6 t Question # Make sure to indicate direction by drawing arrows. Direction is clockwise. dy lim lim t t d t t 5 t t t t t t '.68 y.68.9 and t t ln t t.86 AND Speed '.86 y' y Acceleration ''.86, ''.86.9,.879 To calculate ZERO: To calculate SPEED: To calculate ACCELERATION:

13 TABLE OF VALUES (Calculator) (THE WIRE) Distance (cm) Temperature T C A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T, in degrees Celsius C, of the wire cm from the heated end. The function T is decreasing and twice differentiable.. Estimate T '7. Show the work that leads to your answer. Indicate units of measure.. Write an integral epression in terms of T for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure. 8. Find T ' d and indicate units of measure. Eplain the meaning of T ' temperature of the wire.. Are the data in the table consistent with the assertion that '' Eplain your answer. 8 d in terms of the T for every in the interval 8?

14 . T 8 TABLE OF VALUES (ANSWERS) (THE WIRE) 8 T C/cm C 8 8. T d 8. T d T T ' C The temperature drops 9 C from the heated and of the wire to the other end of the wire. T'' for some value of on the interval 6. No. The MVT guarantees that (See graph of T' below). T T' (estimate) Graph of T' y

15 TABLE OF VALUES (Calculator) (WATER TEMPERATURE) t (days) Wt ( C) The temperature, in degrees Celsius ( C), of the water in a pond is a differentiable function W of time t. The table above shows the water temperature as recorded every days over a 5-day period. ) Use data from the table to find the average change in the water temperature for the 5-day period. ) Use data from the table to find an approimation for W '. Show the computations that lead to your answer. Indicate units of measure. ) Approimate the average temperature, in degrees Celsius, of the water over the time interval t 5 days by using a trapezoidal approimation with subintervals of length t = days. ) A student proposes the function P, given by water in the pond at time t, where t is measured in days and / P t te t, as a model for the temperature of the Pt is measured in degrees Celsius. Find P '. Using appropriate units, eplain the meaning of your answer in terms of water temperature. 5) Use the function P defined in part () to find the average value, in degrees Celsius, of Ptover the time interval t 5 days. 6) Will W' t during the 5-day period? Why or why not?

16 TABLE OF VALUES (Answers) (WATER TEMPERATURE).. W W 5 9 C day W W W W C or: day W 5 9 C or: 5 day W C 9 day C t t C P ' t e te P ' e e / / e day Remember, Calc Derivative: P'.59 or Math 8, y,,.59 This means that the temperature is decreasing at the rate of e 5 t / te dt C 5 t C at t days day Yes. By Mean Value Theorem, W and somewhere t 5, W t Thus, W' somewhere t 5

17 TABLE OF VALUES (Calculator) (PIE PROBLEM). Let yt represent the temperature of a pie that has been removed from a 5 F oven and left to cool in a room with a temperature of 7 F, where y is a differentiable function of t. The table below shows the temperature recorded every five minutes. t (min) yt F A) Use data from the table to find an approimation for y ' 8, and eplain the meaning of ' 8 y in terms of the temperature of the pie. Show the computations that lead to your answer, and indicate units of measure. B) Use data from the table to find the value of y ' t dt, and eplain the meaning of y ' the temperature of the pie. Indicate units of measure. 5 5 t dt in terms of C) A model for the temperature of the pie is given by the function:.6 in minutes and Wt is measured in degrees Fahrenheit Indicate units of measure. W t t 7 8e where t is measured F. Use the model to find the value of ' 8 W. D) Use the model given in part (c) to find the time at which the temperature of the pie is F.

18 TABLE OF VALUES (Answers) (PIE PROBLEM) A) Use data from the table to find an approimation for y ' 8, and eplain the meaning of ' 8 y in terms of the temperature of the pie. Show the computations that lead to your answer, and indicate units of measure F y ' min. When t 8 minutes, the temperature of the pie is decreasing at a rate of approimately 7 F per minute. B) Use data from the table to find the value of y ' t dt, and eplain the meaning of y ' the temperature of the pie. Indicate units of measure. 5 y ' t dt y 5 y F 5 From t minutes to t 5 minutes, the temperature of the pie dropped F. C) A model for the temperature of the pie is given by the function:.6 minutes and Wt is measured in degrees Fahrenheit Indicate units of measure. W t 5 t dt in terms of t 7 8e where t is measure in F. Use the model to find the value of ' 8 W ' F per minute. D) Use the model given in part (c) to find the time at which the temperature of the pie is F. W t when t.9 minutes. W.

19 TABLE OF VALUES (SUGAR MILL). Let y(t) represent the population of the town of Sugar Mill over a -year period, where y is a differentiable function of t. The table below shows the population recorded every two years. t (yrs) 6 8 y people A) Use data from the table to find an approimation for y (7), and eplain the meaning of y (7) in terms of the population of Sugar Mill. Show the computations that lead to your answer. B) Use data from the table to approimate the average population of Sugar Mill over the time interval t by using a left Reimann sum with five equal subintervals. Show the computations that lead to your answer. C) A model for the population of another town, Pine Grove, over the same -year period is given by the function P t the value of P (7). t 5, where t is measured in years and P(t) is measured in people. Use the model to find D) Use the model given in part (c) to find the value of epression in terms of the population of Pine Grove. P t dt. Eplain the meaning of this integral

20 TABLE OF VALUES (Answers) (SUGAR MILL). Let y(t) represent the population of the town of Sugar Mill over a -year period, where y is a differentiable function of t. The table below shows the population recorded every two years. t (yrs) 6 8 y people A) Use data from the table to find an approimation for y (7), and eplain the meaning of y (7) in terms of the population of Sugar Mill. Show the computations that lead to your answer. 85 y ' When t 7 years, the population of Sugar Mill is increasing at a rate of approimately 57.5 people per year. B) Use data from the table to approimate the average population of Sugar Mill over the time interval t By using a left Reimann sum with five equal subintervals. Show the computations that lead to your answer. Average Population y t dt so the average population over the year period was approimately 8. people. C) A model for the population of another town, Pine Grove, over the same -year period is given by the P t function the value of P (7). t 5, where t is measured in years and P(t) is measured in people. Use the model to find P ' 7 56 people per year D) Use the model given in part (c) to find the value of epression in terms of the population of Pine Grove. P t dt P t dt. Eplain the meaning of this integral 6. people. This means that the average population of Pine Grove over the year period was approimately 6. people.

21 TABLE OF VALUES (BOWL OF SOUP). A bowl of soup is place on the kitchen counter to cool. Let T() represent the temperature of the soup at time, where T is a differentiable function of. The temperature of the soup at selected times is given in the table below. (min) 7 T() F A) Use data from the table to find: T ' d Eplain the meaning of this definite integral in terms of the temperature of the soup. B) Use data from the table to find the average rate of change of T over the time interval to 7 C) Eplain the meaning of: T d In terms of the temperature of the soup, and approimate the value of this integral epression by using a trapezoidal sum with three subintervals

22 TABLE OF VALUES (ANSWERS) (BOWL OF SOUP). A bowl of soup is place on the kitchen counter to cool. Let T() represent the temperature of the soup at time, where T is a differentiable function of. The temperature of the soup at selected times is given in the table below. (min) 7 T() F A) Use data from the table to find: T ' d Eplain the meaning of this definite integral in terms of the temperature of the soup. T ' d T T 6 F From to minutes, the temperature of the soup dropped F B) Use data from the table to find the average rate of change of T over the time interval to 7 Average rate of change = T 7 T 99 7 F / min C) Eplain the meaning of: T d In terms of the temperature of the soup, and approimate the value of this integral epression by using a trapezoidal sum with three subintervals T d represents the average temperature of the soup over the -minute period and is approimately equal to: F

23 TABLE OF VALUES (WATER INTO A TANK). The rate at which water is being pumped into a tank is given by the continuous, increasing function Rt. A table of values of Rt, for the time interval t minutes, is shown below t (min) 9 7 Rt (gal/min) A) Use a right Riemann sum with four subintervals to approimate the value of: R t dt Is your approimation greater or less than the true value? Give a reason for your answer. B) A model for the rate at which water is being pumped into the tank is given by the function: t W t 5e where t is measured in minutes and. Wt is measured in gallons per minute. Use the model to find the average rate at which water is being pumped into the tank from t to t minutes. C) The tank contained gallons of water at time t. Use the model given in part (b) to find the amount of water in the tank at t = minutes

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25 TABLE OF VALUES (ANSWERS) (WATER INTO A TANK). The rate at which water is being pumped into a tank is given by the continuous, increasing function Rt. A table of values of Rt, for the interval t minutes, is shown below t (min) 9 7 Rt (gal/min) A) Use a right Riemann sum with four subintervals to approimate the value of: R t dt Is your approimation greater or less than the true value? Give a reason for your answer. R t dt gallons Since R is positive, this is an estimate of the amount of water pumped into the tank during the -minute period. Since R increases on t, the right Riemann sum approimation of 75 gallons is greater than R t dt. B) A model for the rate at which water is being pumped into the tank is given by the function: t W t 5e Where t is measured in minutes and. Wt is measured in gallons per minute. Use the model to find the average rate at which water is being pumped into the tank from t to t minutes. Average Rate = W t dt.55 gal/min C) The tank contained gallons of water at time t. Use the model given in part (b) to find the amount of water in the tank at t = minutes W t dt gallons

26 5. Car A has positive velocity v TABLE OF VALUES (CAR VELOCITY) A t as it travels on a straight road, where va is a differentiable function of t. The velocity is recorded for selected values over the time interval t seconds, as shown in the table below. t (sec) 5 7 v t A (ft/sec) A) Use data from the table to approimate the acceleration of Car A at t = 8 seconds. Indicate units of measure. B) Use data from the table to approimate the distance traveled by Car A over the interval t seconds by using a trapezoidal sum with four subintervals. Show the computations that lead to your answer, and indicate units of measure. C) Car B travels along the same road with an acceleration of a t t ft /sec. At time t = seconds, the velocity of car B is ft/sec. Which car is traveling faster at time t = 7 seconds? Eplain your answer. B

27 5. Car A has positive velocity v TABLE OF VALUES (Answers) (CAR VELOCITY) A t as it travels on a straight road, where va is a differentiable function of t. The velocity is recorded for selected values over the time interval t seconds, as shown in the table below. t (sec) 5 7 v t A (ft/sec) A) Use data from the table to approimate the acceleration of Car A at t = 8 seconds. Indicate units of measure. a Let A aa t be the acceleration of Car A at time t. Then: va va ft / sec ft sec sec B) Use data from the table to approimate the distance traveled by Car A over the interval t seconds by using a trapezoidal sum with four subintervals. Show the computations that lead to your answer, and indicate units of measure. Distance: v dt A ft C) Car B travels along the same road with an acceleration of a t t ft /sec. At time t = seconds, the velocity of car B is ft/sec. Which car is traveling faster at time t = 7 seconds? Eplain your answer. B Let be the velocity of Car B at time t. Then: v t t dt t² t C At t, we have 9 6 C, so that C and v t t² t. B Hence, v v 7. We conclude that Car A is traveling faster at time t 7 seconds. B A B

28 RATES OF CHANGE PROBLEM # (CALCULATOR) Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by: t F t 8 sin for t Where F t is measured in cars per minute and t is measured in minutes.. To the nearest whole number, how many cars pass through the intersection over the minute period?. Is the traffic flow increasing or decreasing at t 5? Justify.. What is the average value of the traffic flow over the time interval t 7? Indicate units of measure.. What is the average rate of change of the traffic flow over the time interval t 7? Indicate units of measure. 5. At what time, t, is the traffic flow the greatest? What is the greatest flow? PROBLEM # (CALCULATOR) A water tank at Camp Diamond Bar holds gallons of water at time t. During the time interval t hours, water is pumped into the tank at the rate: W t t 6 95 t sin gallons per hour. During the same time interval, water is removed from the tank at the rate 75sin Rt t gallons per hour.. Is the amount of water in the tank increasing at time t 5? Why or why not?. To the nearest whole number, how many gallons of water are in the tank at time t?. At what time, t, for t, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.. For t, no water is pumped into the tank, but water continues to be removed at the rate Rt until the tank become empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral epression that can be used to find the value of k. 5. What is the average rate of change in the amount of water in tank for t hours?

29 . F( t) dt 86 cars. Decreasing. Because F F.. PROBLEM # ANSWERS: ' 5 ' F t dt cars/minute F 7 F F '( t) dt.8 cars/minute 5. F() = 8, F() = 78.6, and F( ) = 86 so greatest at t minutes, 86 cars per minute.. No. the amount of water is not increasing at 5 PROBLEM # ANSWERS: t because W R. 6 gallons W t Rt dt 6.6. At t 6.95 Because: at t, there are gallons of water in the tank at t 6.95, there are 55. gallons of water in the tank. at t, there are 6.6 gallons of water in the tank. k. 5. R t dt W t R t dt ( ( ) ( )) 6.97 gallons/hour

30 ANALYZING A PARTICLE PROBLEM PROBLEM # (NO CALCULATOR) A particle moves along the ais with the velocity at time. Find the acceleration of the particle at time t. v t e t. t given by. Is the speed of the particle increasing at time t? Give a reason for your answer.. Find all values of t at which the particle changes direction. Justify your answer.. What is the average velocity of the particle over the interval t? 5. Find the total distance traveled by the particle over the interval t? PROBLEM # (CALCULATOR) A particle moves along the y ais so that its velocity v at time y. Find the acceleration of the particle at time t. t t is given by vt tan e and. Is the speed of the particle increasing or decreasing at time t? Give a reason for your answer.. Find the time t at which the particle reaches its highest point. Justify your answer.. Find the position of the particle at time t. Is the particle moving toward the origin or away from the origin at time t? Justify your answer. 5. Find the total distance traveled by the particle over the interval t?

31 t. '. v e a a t v t e a e e PROBLEM # ANSWERS: and Speed is increasing. Particle changes direction at t v t when e, so t. t because vt changes from positive to negative at t. t t AV e dt t e e e e e 5. t t t t TD e dt e dt t e t e e e e e e e e e PROBLEM # ANSWERS:. a v'.. v.6 Speed is increasing since a and. vt when t.. yt must have a maimum at. t because ' Maimum doesn t occur at endpoints because when t... y v t dt.6 5. vt TD dt.9 v are both negative. y t changes from positive to negative at t.. yt increases on the interval (,.) and. The particle is moving away from the origin since yt decreases v and y.

32 Question # (Calculator) EULER S METHOD/APPROXIMATION Let f be the function whose graph goes through the point,6 and whose e derivative is given by f '. Write an equation of the line tangent to the graph of f at and use it to approimate.. Use Euler s method, starting at with a step size of.5, to approimate f... Is this approimation (from #) less than or greater than f.? Why?.. Use f d f ' to evaluate.. Question # (No Calculator) Consider the differential equation given by: dy y d. On the aes provided below, sketch a slope field for the given differential equation. y f. f.. Let y f be the particular solution to the given differential equation with the initial condition Use Euler s method starting at, with a step size of., to approimate f.. Show work!. Find the particular solution y f to the given differential equation with the initial condition Use your solution to find f. Question # (No Calculator) f. Let f be the function satisfying f ' f, for all real numbers, with f and f. Evaluate f d. Show the work that lead to your answer.. Use Euler s method, starting at with a step size of.5, to approimate. Write an epression for y f f. f. lim. dy by solving the differential equation y with the initial condition d

33 EULER S METHOD/APPROXIMATION ANSWERS Question # e e e f '. y 6 f f.5 f f ' f f. 6.7 ' e e e f '' For, f '' so the graph of f is concave upward on,.. the Euler approimation lines at and.5 lie below the graph. Under approimates... f ' d f. f f. f f ' d f Question # y f. f f '... f. f. f '....5 dy y dy d ln y C y C e C e C y e d y..... f e e e / / Question # f d f ' d lim f ' d lim f lim f b f b b b f.5 f f '.5.5 f b f ' dy d ln y k y C e C e C e y e e y / / / / / b

34 IMPLICIT DIFFERENTIATION PROBLEM # (NO CALCULATOR) Consider the curve y y. Use implicit differentiation to show that dy y d. Find the equation of all horizontal tangent lines.. Find the equation of all vertical tangent lines.. Find the equation of the tangent line(s) at. 5. Using the tangent line at, approimate. 6. Is the curve increasing or decreasing at 7. Is the curve concave up or down at y.? Justify your answer.? Justify your answer. 8. Would a tangent line approimation overestimate or underestimate at? Why? PROBLEM # (NO CALCULATOR) Consider the curve y y. Use implicit differentiation to show that dy y y d y. Find the equation of all horizontal tangent lines.. Find the equation of all vertical tangent lines.. Is the curve increasing or decreasing at (, )? Justify your answer. 5. Is the curve concave up or down at (, )? Justify your answer.

35 PROBLEM # ANSWERS: 7. y {This occurs when } or dy 6. Decreasing because 6 d at. y d y 7. Concave up because 8 d at 8. Underestimate because curve is concave up at... PROBLEM # ANSWERS: dy dy dy y y y y y y d d d dy dy y y y y y y y d d y y dy y y y y y or y y or y d y If y, then y y BAD!!!!! If y, then If y, then y y y y y y y y or y dy y y y or y or y d y If, then y y y y BAD!!!!! If y, then y y y y y y y y y If y, then y. Increasing because 5. y or dy y y dy d y d dy y y dy, d y d ' y y, dy ' y, dy y y, d d ' dy y ', d ' ' d y d y Concave Down d 9 d

36 LOGISTIC GROWTH Question # A certain rumor spreads through a community at the rate of y y population that has heard the rumor at time t. dy where y is the proportion of the dt. What proportion of the population has heard the rumor when it is spreading the fastest?. If at time t ten percent of the people have heard the rumor, find y as a function of t.. At what time t is the rumor spreading the fastest?

37 LOGISTIC GROWTH (ANSWERS) y is growing the fastest when y one-half of the carrying capacity, A. So y = A () proportion when rumor is spreading the fastest is y Logistic growth differential equation is dy dt ky A y and in this problem dy dt So A and k. The solution is y Since y. when t. So y 9e t Ce Akt So in this problem y Ce C From #, the rumor spreads the fastest when y. Ce t..c C 9 y y 9e t e t 9e t 9 t ln9 t ln9 Leave the answer in this form if it is a free response problem, but it does simplify to ln.

38 GRAPHING PROBLEM # (NO CALCULATOR) (e) Set up, but do not evaluate, an epression that would result in the area of f () for.

39 PROBLEM # ANSWERS: (e) Area = f d f d f d